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import OAI.Analysis.HardSphere.FormDomain.Basic
import OAI.Analysis.HardSphere.FormDomain.Energy
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
namespace HardSphere
theorem IsGroundVector.first_variation {N : ℕ} {L : Length} {a : ℝ}
{u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L)
(hv : v ∈ dirichletDomain N L a) (hb : bosonic v.1) :
(∑ p : Direction N, inner ℂ (u.2 p) (v.2 p)) =
(energy u : ℂ) * inner ℂ u.1 v.1 := by
apply Variational.ground_pairing (groundFormSubmodule N L a) u
· exact mem_groundFormSubmodule.mpr ⟨hu.1, hu.2.1⟩
· exact hu.2.2.1
· intro w hw hn
exact hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1
(mem_groundFormSubmodule.mp hw).2 hn
· exact mem_groundFormSubmodule.mpr ⟨hv, hb⟩
theorem IsGroundVector.rayleigh {N : ℕ} {L : Length} {a : ℝ}
{u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L)
(hv : v ∈ dirichletDomain N L a) (hb : bosonic v.1) :
energy u * ‖v.1‖ ^ 2 ≤ energy v := by
exact Variational.rayleigh (groundFormSubmodule N L a) u
(fun w hw hn ↦ hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1
(mem_groundFormSubmodule.mp hw).2 hn) v
(mem_groundFormSubmodule.mpr ⟨hv, hb⟩)
@[simp] lemma torusDistance_comm (L : Length) (x y : Torus L) :
torusDistance L x y = torusDistance L y x := by
simp only [torusDistance, dist_comm]
lemma allowed_iff_ne {N : ℕ} {L : Length} {a : ℝ} {X : Configuration N L} :
X ∈ allowed N L a ↔ ∀ i j, i ≠ j → a < torusDistance L (X i) (X j) := by
constructor
· intro h i j hij
rcases lt_or_gt_of_ne hij with hlt | hgt
· exact h i j hlt
· simpa only [torusDistance_comm] using h j i hgt
· intro h i j hij
exact h i j (ne_of_lt hij)
lemma allowed_permute {N : ℕ} {L : Length} {a : ℝ}
(σ : Equiv.Perm (Fin N)) (X : Configuration N L) :
permuteConfiguration σ X ∈ allowed N L a ↔ X ∈ allowed N L a := by
simp only [allowed_iff_ne, permuteConfiguration]
constructor
· intro h i j hij
simpa using h (σ.symm i) (σ.symm j) (σ.symm.injective.ne hij)
· intro h i j hij
exact h (σ i) (σ j) (σ.injective.ne hij)
def permuteLift {N : ℕ} (σ : Equiv.Perm (Fin N)) :
LiftedConfiguration N ≃L[ℝ] LiftedConfiguration N where
toFun X := fun i ↦ X (σ i)
invFun X := fun i ↦ X (σ.symm i)
left_inv X := by funext i; simp
right_inv X := by funext i; simp
map_add' _ _ := rfl
map_smul' _ _ := rfl
continuous_toFun := continuous_pi fun _ ↦ continuous_apply _
continuous_invFun := continuous_pi fun _ ↦ continuous_apply _
lemma permuteLift_direction {N : ℕ} (σ : Equiv.Perm (Fin N)) (p : Direction N) :
permuteLift σ (Pi.single p.1 (Pi.single p.2 1)) =
Pi.single (σ.symm p.1) (Pi.single p.2 1) := by
ext i j
change (Pi.single p.1 (Pi.single p.2 (1 : ℝ)) : LiftedConfiguration N) (σ i) j = _
by_cases h : i = σ.symm p.1
· subst i; simp
· have h' : σ i ≠ p.1 := fun hh ↦ h ((Equiv.eq_symm_apply σ).mpr hh)
simp [Pi.single_eq_of_ne h, Pi.single_eq_of_ne h']
lemma coordDerivative_permute {N : ℕ} {L : Length} {a : ℝ}
(f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) (p : Direction N) :
coordDerivative N L (f.toFun ∘ permuteConfiguration σ) p =
coordDerivative N L f.toFun (σ.symm p.1, p.2) ∘ permuteConfiguration σ := by
funext X
change (fderiv ℝ ((fun Y ↦ f.toFun (project N L Y)) ∘ permuteLift σ)
(representative N L X)) (Pi.single p.1 (Pi.single p.2 1)) = _
rw [fderiv_comp _ (f.smooth.differentiable (by simp)).differentiableAt
(permuteLift σ).differentiableAt, (permuteLift σ).fderiv]
simp only [ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe,
permuteLift_direction]
rfl
def SmoothTest.permute {N : ℕ} {L : Length} {a : ℝ}
(f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) : SmoothTest N L a where
toFun := f.toFun ∘ permuteConfiguration σ
smooth := f.smooth.comp (permuteLift σ).contDiff
compact := HasCompactSupport.of_compactSpace _
support_allowed := by
intro X hX
apply (allowed_permute σ X).mp
exact f.support_allowed
(tsupport_comp_subset_preimage (f := permuteConfiguration σ) f.toFun (continuous_pi fun i ↦ continuous_apply (σ i)) hX)
memLp := f.memLp.comp_measurePreserving (measurePreserving_permute σ)
partial_memLp := by
intro p
rw [coordDerivative_permute]
exact (f.partial_memLp (σ.symm p.1, p.2)).comp_measurePreserving
(measurePreserving_permute σ)
section
attribute [local instance] scalarL2_addCommMonoid scalarL2_module
lemma continuous_permutationPullback_jet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
Continuous (fun u : Jet N L ↦ (permutationPullback σ u.1,
fun direction : Direction N ↦
permutationPullback σ (u.2 (σ.symm direction.1, direction.2)))) := by
let pullback : ScalarL2 N L →ₗᵢ[ℂ] ScalarL2 N L :=
Lp.compMeasurePreservingₗᵢ ℂ (permuteConfiguration σ) (measurePreserving_permute σ)
exact (pullback.continuous.comp continuous_fst).prodMk
(continuous_pi fun p ↦ pullback.continuous.comp ((continuous_apply _).comp continuous_snd))
def permutationJet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
Jet N L →L[ℂ] Jet N L where
toFun u := (permutationPullback σ u.1,
fun p ↦ permutationPullback σ (u.2 (σ.symm p.1, p.2)))
map_add' _ _ := by simp [map_add]; rfl
map_smul' _ _ := by simp [map_smul]; rfl
cont := continuous_permutationPullback_jet σ
end
lemma SmoothTest.jet_permute {N : ℕ} {L : Length} {a : ℝ}
(f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) :
(f.permute σ).jet = permutationJet σ f.jet := by
apply Prod.ext
· apply Lp.ext
exact (MemLp.coeFn_toLp (f.permute σ).memLp).trans
((Lp.coeFn_compMeasurePreserving f.jet.1 (measurePreserving_permute σ)).trans
((measurePreserving_permute σ).quasiMeasurePreserving.ae_eq_comp
(MemLp.coeFn_toLp f.memLp))).symm
· funext p
apply Lp.ext
have h := (measurePreserving_permute (L := L) σ).quasiMeasurePreserving.ae_eq_comp
(MemLp.coeFn_toLp (f.partial_memLp (σ.symm p.1, p.2)))
refine (MemLp.coeFn_toLp ((f.permute σ).partial_memLp p)).trans ?_
change coordDerivative N L (f.toFun ∘ permuteConfiguration σ) p =ᵐ[volume] _
rw [coordDerivative_permute]
exact ((Lp.coeFn_compMeasurePreserving _ (measurePreserving_permute σ)).trans h).symm
lemma permutationJet_mem_domain {N : ℕ} {L : Length} {a : ℝ}
(σ : Equiv.Perm (Fin N)) {u : Jet N L} (hu : u ∈ dirichletDomain N L a) :
permutationJet σ u ∈ dirichletDomain N L a := by
apply map_mem_closure (permutationJet σ).continuous hu
rintro _ ⟨f, rfl⟩
exact ⟨f.permute σ, f.jet_permute σ⟩
lemma energy_permutationJet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N))
(u : Jet N L) : energy (permutationJet σ u) = energy u := by
change (∑ p : Direction N, ‖Lp.compMeasurePreserving _ (measurePreserving_permute σ)
(u.2 (σ.symm p.1, p.2))‖ ^ 2) = _
simp only [Lp.norm_compMeasurePreserving]
exact Equiv.sum_comp (Equiv.prodCongr σ.symm (Equiv.refl (Fin 3)))
(fun p ↦ ‖u.2 p‖ ^ 2)
end HardSphere
end
end OAI
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